Write a linear equation from a slope and a point
If you know a line's slope and just one point it passes through, you have enough information to write its equation. This is a powerful tool for modeling linear relationships in the real world.
Do this: Read the concept below, then try the quiz or activity.
Concept
The most common form of a linear equation is the slope-intercept form: y = mx + b. * m is the slope. * b is the y-intercept (the point where the line crosses the y-axis).
To write an equation from a slope and a point, you need to find the value of 'b'.
The Plug-in Method
1. Start with the slope-intercept formula: y = mx + b. 2. Plug in the slope (m) and the coordinates of the given point (x, y). 3. Solve the resulting equation for 'b'. 4. Write the final equation with the values for m and b.
Example: Write the equation for a line that has a slope of 2 and passes through the point (3, 7).
1. Start with the formula: y = mx + b
2. Plug in the values: We know m = 2, x = 3, and y = 7.
* 7 = 2(3) + b
3. Solve for b:
* 7 = 6 + b
* Subtract 6 from both sides: 7 - 6 = b
* b = 1
4. Write the final equation: Now we know m = 2 and b = 1.
* y = 2x + 1Point-Slope Form (An Alternative Method) Another useful form of a linear equation is the point-slope form: y - y₁ = m(x - x₁) * You just plug in the slope 'm' and the point (x₁, y₁).
Example (Same as above): slope = 2, point = (3, 7)
1. Start with the formula: y - y₁ = m(x - x₁)
2. Plug in the values: m = 2, x₁ = 3, y₁ = 7.
* y - 7 = 2(x - 3)
3. (Optional) Convert to slope-intercept form:
* Distribute the 2: y - 7 = 2x - 6
* Add 7 to both sides: y = 2x + 1Both methods give the same result!
Key Idea: Every point on a line is a solution to its equation. That's why you can plug in the x and y from any point on the line to help you find the missing piece of the equation (the y-intercept 'b').
Try it
Practice: Write a linear equation from a slope and a point.